Isn't that what a personal liability policy covers? Some apartments I have been in required you to have one.

AFAIK personal liability is something like when someone comes to your house and slips on a puddle you have on the floor and hurts themselves and then sues you. It's not about damage you did yourself, on purpose, to the property you rent from someone else or someone else does on purpose against a property you rent out.

All the insurance policies were about accidental damage on either side.

Then why would an apartment require you to carry it and list them as the interested party? What would they care if it just protects you against drunk partners at your place that get injured?

Your insurance as the property owner can cover vandalism, theft, and other intentional damage if you choose (and you should for all the reasons listed in this thread).

When I tried shopping online for it as a research project, it explicitly excluded intentional damage in the fine print, and there was no extra option to purchase it separately. Could someone show me a policy that has that explicitly, or at the very least does not exclude it explicitly like I found?

I'm not going to share the details of my own policies, but every major insurer offered it to me without asking. Maybe it's state or region dependant.

Or are you talking about intentional damage by the owner?

More intentional damage by the renter for the landlord's policy. Usually the exception is under the wording of 'vandalism'. I think intentional damage by someone who bought the place is an OK thing to have an exception for. Although I wonder if banks have insurance for that case.